The present invention relates generally to a tensor decomposition method, and more particularly, but not by way of limitation, to a system, method, and computer program product for optimal multi-dimensional data compression by tensor-tensor decompositions.
Most of real-world data is inherently multidimensional. Many operators and models are natively multi-way. The use of higher-order tensor representations (i.e., multi-way array representations) has become ubiquitous in science and engineering applications. It is often natural to store the data according to the variable labelling to which it relates (i.e., vertical, horizontal and depth dimensions might correspond to variables in two spatial dimensions and a time dimension). It has become clear that processing data in tensor format through the use of certain known decompositions relevant to the specific application can capture correlations or patterns in data that are not obvious when the data is treated in a matrix format.
Moreover, structural redundancies can be captured via tensor decompositions in ways that allow for better compression of data. However, first, such data needs to be presented in a tensor format in which this can be revealed. Consider, for example, a single vector νϵ (i.e., a 1st order tensor) which is actually the Kronecker product of two vectors, one of length n and one of length m. In order to reveal the two vectors in the Kronecker form, one needs to reshape the data to an m×n or n×m matrix (i.e., a 2nd order tensor), and observe that it is a rank-one matrix. From the rank-1 matrix factors, one obtains a pair of vectors needed to form v. Thus, the implicit storage cost of v is only (m+n), rather than mn, a great savings if m and n are large.